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Chapter 16: Problem 7

Sketch the following polar rectangles. $$R=\\{(r, \theta): 0 \leq r \leq 5,0 \leq \theta \leq \pi / 2\\}$$

Short Answer

Expert verified
Question: Sketch the polar rectangle formed by the polar coordinates $(r, \theta): 0 \leq r \leq 5,0 \leq \theta \leq \pi / 2$ and describe its properties. Answer: The polar rectangle is formed by the radial distance between 0 and 5 units, inclusive, and the angle between 0 and $\pi / 2$ (90 degrees), inclusive. It lies within the first quadrant of the plane, and its shape is a right triangle with one vertex at the origin, one on the positive x-axis at $(5, 0)$, and one on the positive y-axis at $(5, \pi / 2)$. The hypotenuse of the right triangle is the circle with radius 5, and the legs are the lines representing the limits of the angle, $\theta = 0$ and $\theta = \pi / 2$.

Step by step solution

01

Analyze the given limits of the polar coordinates

Here, we have two ranges of values for the polar coordinates: \(0 \leq r \leq 5\) and \(0 \leq \theta \leq \pi / 2\). The limit \(0 \leq r \leq 5\) restricts the radial distance (i.e., the distance from the origin) to be between 0 and 5 units, inclusive. The limit \(0 \leq \theta \leq \pi / 2\) restricts the angle to be between 0 and \(\pi / 2\) (90 degrees), inclusive.
02

Plot the points corresponding to the limits of the radial distance

Now, we'll plot the points corresponding to the limits of the radial distance. Since the range of \(r\) is \(0 \leq r \leq 5\), this means that the polar rectangle will have a radius of 5 units. To represent this, draw a circle centered at the origin with a radius of 5 units.
03

Plot the points corresponding to the limits of the angle

The range of the angle \(\theta\) is \(0 \leq \theta \leq \pi / 2\). This restricts the polar rectangle to lie within the first quadrant of the plane, as \(0 \leq \theta \leq \pi / 2\) corresponds to angles between 0 and 90 degrees. To represent this, draw two rays in the first quadrant: one along the positive x-axis (i.e., \(\theta = 0\)), and one along the positive y-axis (i.e., \(\theta = \pi / 2\)).
04

Draw the polar rectangle

Now that we have plotted the points corresponding to the limits of the radial distance and angle, we can sketch the polar rectangle. To do this, draw a line connecting the two points where the circle (from Step 2) intersects the rays (from Step 3). This will form a right triangle, where the hypotenuse is the circle with radius 5 (i.e., \(r = 5\)) and the legs are the lines representing the limits of the angle (i.e., \(\theta = 0\) and \(\theta = \pi / 2\)). The polar rectangle has now been sketched.

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